 where \(\gamma ((\mathbf {Q}^*)^k \mathbf {Q}^k)\)  denotes the spectral gap of the self-adjoint operator \((\mathbf {Q}^*)^k \mathbf {Q}^k\) . It is known that the pseudo-spectral gap \(\gamma _{\rm {ps}}\)  and the mixing time \(t_{\rm {mix}}\)  of an ergodic (irreducible) and reversible Markov chain is related to each other [1]}. In [2]}, the authors provided algorithms to estimate \(t_{\rm {mix}}\)  and \(\gamma _{\rm {ps}}\)  from a single trajectory.
